Delay-induced dynamics in a nonlinear crime interaction model with periodic forcing

Abstract

A nonlinear time-delay model is proposed to describe the interaction dynamics between criminal and non-criminal populations, combining social influence mechanisms, saturation effects represented by a Holling type II functional response, and time-dependent law-enforcement actions. The delay accounts for the latency between exposure to criminal behavior and behavioral response, introducing memory effects that naturally lead to a delay differential equations framework. Fundamental analytical properties, including positivity, global existence, and invariance of the feasible region, are established to ensure the mathematical consistency of the population interpretation. In the autonomous setting, explicit threshold conditions governing the stability of the criminal-free equilibrium and the emergence of coexistence states are derived, while the delay is shown to induce stability switches and oscillatory regimes through characteristic root crossings. In the non-autonomous case, topological degree arguments guaranty the existence of strictly positive periodic solutions, indicating that long-term dynamics depend primarily on the averaged law enforcement intensity measures rather than on short-term fluctuations. These results identify time delay as a key structural mechanism underlying recurrent patterns and complex temporal behavior in crime dynamics.